%5E2-in-standard-form.jpeg "(3-2i)^2 In Standard Form")
Simplifying (3 - 2i)² to Standard Form
In mathematics, standard form for complex numbers is expressed as a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1. To express the complex number (3 - 2i)² in standard form, we need to expand and simplify the expression.
1. Expanding the Expression:
We can expand the expression using the distributive property or by recognizing it as a perfect square:
- Using Distributive Property:
(3 - 2i)² = (3 - 2i)(3 - 2i) = 3(3 - 2i) - 2i(3 - 2i) = 9 - 6i - 6i + 4i²
- Recognizing Perfect Square:
(3 - 2i)² = (3)² - 2(3)(2i) + (2i)² = 9 - 12i + 4i²
2. Simplifying with the Value of i²:
Since i² = -1, we can substitute it into the expanded expression:
9 - 6i - 6i + 4i² = 9 - 6i - 6i + 4(-1) = 9 - 6i - 6i - 4
3. Combining Real and Imaginary Terms:
Now, combine the real terms (9 and -4) and the imaginary terms (-6i and -6i):
(9 - 4) + (-6i - 6i) = 5 - 12i
Therefore, (3 - 2i)² in standard form is 5 - 12i.